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This a sketch of the proof, the details can be found here. I will offer this sketch because that paper was not intended to prove this result in particular, and I think that a proof might have been written somewhere else. Consider Mills ratio defined by: $$\varphi(x)=e^{x^2/2}\int_x^\infty e^{-t^2/2}dt.$$ Proposition 1. There is a unique sequence of ...

1

Corrected posting: ${\rm PSL}(2,\mathbb{Z})$ is the free product of $\langle S \rangle$ and $\langle ST \rangle,$ so, setting $U = ST$, for a given matrix $M$ (mod scalars), the expression for $M$ is unique if we write it in the form $S^{\alpha} U^{b_{1}}SU^{b_{2}}SU^{b_{3}}\ldots SU^{b_{n}}S^{\beta}$, where each $b_{i} \in \{1,-1\}$ (or $\{1,2 \}$ if you ...

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I believe there is no presently known "reasonably explicit" description (i.e. an $n$-term recursion formulation for some positive integer $n)$ of continued fraction expansions that can be used to characterize even just the cubic irrationalities, although a recent paper by Nadir Murru---On the Hermite problem for cubic irrationalities---appears to give such a ...

2

If $n$ is finite: $n\in\mathbb{Q}$ This is because a finite continued fraction is the result of a finite amount of additions and divisions of integers. If $n$ is infinite and periodic $n$ is an irrational root of $ax^2+bx+c=0$ where $a,b,c\in\mathbb{Q}$ If $n$ is infinite and non-periodic $n$ is not expressible as a root of a second degree or lower ...

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